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The general eccentric connectivity index of a graph R is defined as ξ e c (R) = ∑ u ∈ V (G) d (u) e c (u) α , where α is any real number, e c (u) and d (u) represent the eccentricity and the degree of the vertex u in R, respectively. In this paper, some bounds on the general eccentric connectivity index are proposed in terms of graph-theoretic parameters, namely, order, radius, independence number, eccentricity, pendent vertices and cut edges. Moreover, extremal graphs are characterized by these bounds. [ABSTRACT FROM AUTHOR]

The first and second Zagreb indices of a graph G are defined as: M 1 (G) = ∑ v ∈ V (G) d (v) 2 and M 2 (G) = ∑ u v ∈ E (G) d (u) d (v) , where d (v) is the degree of the vertex v. A graph G is said to be a quasi unicyclic graph, if there exists a vertex z ∈ V (G) , such that G − z is a unicyclic graph and z is called a quasi vertex. For any integer k ≥ 1 a graph G is called a k -generalized quasi unicyclic graph, if there exists a subset V k ⊆ V (G) with cardinality k such that G − V k is a unicyclic graph but for every subset V k − 1 of cardinality k − 1 of V (G) , the graph G − V k − 1 is not unicyclic graph. In this paper, we investigate the upper and lower bounds on first and second Zagreb indices for k -generalized quasi unicyclic graphs. Moreover, we characterize the extremal graphs with maximum and minimum Zagreb indices. [ABSTRACT FROM AUTHOR]

A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V (G) with cardinality ℓ such that G − A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for ℓ = 1 and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs. [ABSTRACT FROM AUTHOR]

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D (G) = ∑ u ∈ V (G) d (u) − 1 where d (u) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ (G) = ∑ u ∈ V (G) d (u) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified. [ABSTRACT FROM AUTHOR]